
And thus \( \frac{\partial f^{\beta _t}}{\partial P_{ijk}}(P^{(t-1)}) = \tilde{\mu }_{ij}\times {b}_k\) .
The result then follows from [1]} and is a consequence of the rearrangement inequality [2]}: \(Q^{(t)}\)  is obtained by sorting \(\tilde{\mu }_{ij}\)  in increasing order, or equivalently, by sorting \(-\tilde{\mu }_{ij}\)  in decreasing order.
